Multi-scale Geometric Modeling of Ambiguous Shapes with : oleranced Balls and Compoundly Weighted alpha-shapes.

Computer Graphics Forum (Impact Factor: 1.64). 01/2010; 29:1713-1722. DOI: 10.1111/j.1467-8659.2010.01780.x
Source: DBLP

ABSTRACT Dealing with ambiguous data is a challenge in Science in general and geometry processing in particular. One route of choice to extract information from such data consists of replacing the ambiguous input by a continuum, typically a one-parameter family, so as to mine stable geometric and topological features within this family. This work follows this spirit and introduces a novel framework to handle 3D ambiguous geometric data which are naturally modeled by balls. First, we introduce {\em toleranced balls} to model ambiguous geometric objects. A toleranced ball consists of two concentric balls, and interpolating between their radii provides a way to explore a range of possible geometries. We propose to model an ambiguous shape by a collection of toleranced balls, and show that the aforementioned radius interpolation is tantamount to the growth process associated with an additively-multiplicatively weighted Voronoi diagram (also called compoundly weighted or CW). Second and third, we investigate properties of the CW diagram and the associated CW $\alpha$-complex, which provides a filtration called the $\lambda$-complex. Fourth, we propose a naive algorithm to compute the CW VD. Finally, we use the $\lambda$-complex to assess the quality of models of large protein assemblies, as these models inherently feature ambiguities.

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    ABSTRACT: The Voronoi diagram is a geometric object which is widely used in many areas. Recently it has been shown that under mild conditions Voronoi diagrams have a certain continuity property: small perturbations of the sites yield small perturbations in the shapes of the corresponding Voronoi cells. However, this result is based on the assumption that the ambient normed space is uniformly convex. Unfortunately, simple counterexamples show that if uniform convexity is removed, then instability can occur. Since Voronoi diagrams in normed spaces which are not uniformly convex do appear in theory and practice, e.g., in the plane with the Manhattan (ell_1) distance, it is natural to ask whether the stability property can be generalized to them, perhaps under additional assumptions. This paper shows that this is indeed the case assuming the unit sphere of the space has a certain (non-exotic) structure and the sites satisfy a certain "general position" condition related to it. The condition on the unit sphere is that it can be decomposed into at most one "rotund part" and at most finitely many non-degenerate convex parts. Along the way certain topological properties of Votonoi cells (e.g., that the induced bisectors are not "fat") are proved.

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